__TECHNICAL FIELD__
The present invention relates to a digital filter used
in a digital signal processing system.

__BACKGROUND ART__
These days, digital signal processing systems employed
in the fields of audio signal processing, image processing, telecommunications,
automatic control and the like have brought widespread use of digital filters which
are capable of performing intended signal processing by digital operations.

On the other hand, attention has been focused on such techniques
as the oversampling technique for allowing signals being processed to be sampled
at a frequency higher than the Nyquist frequency, the noise shaping technique for
reducing quantization noise (quantumized error) or the like, and the Delta-Sigma
modulation scheme (also referred to as the Sigma-Delta modulation scheme). Those
digital filters that employ these digital signal processing technologies have been
suggested.

For example, in the field of digital audio, in a non-patent
literature suggested is a digital audio filter, configured as shown in Fig. 1, which
employs these oversampling technique, noise shaping technique, and Delta-Sigma modulation
scheme.

As shown in Fig. 2A the digital audio filter 1 has an improved
configuration based on the configuration of a typical digital audio filter which
includes an audio filter section 2 for applying an intended filter characteristic
to an input signal X to adjust gain and phase characteristics or the like; an adder
3; a noise shaping section 4 made up of a low-pass filter and a band-pass filter
which are adapted for the audio band; and a quantizer section 5. The digital audio
filter 1 receives a one-bit stream digital audio signal of PWM (Pulse Width Modulation)
modulated waves or PDM (Pulse Density Modulation) modulated waves, which has been
oversampled at a frequency higher than the Nyquist frequency, as an input signal
(hereinafter referred to as the "one-bit input signal") X. The digital audio filter
1 also performs predetermined digital filtering operations on the one-bit input
signal X and then re-quantizes it into a one-bit stream digital audio signal to
output the re-quantized signal as a one-bit output signal Y.

That is, the digital audio filter 1 shown in Fig. 1 can
be configured as an nth order digital filter (n is an arbitrary natural number),
and has been suggested, by way of example, as a fifth order digital filter. The
digital audio filter 1 is further configured such that integrators IG0 to IG4 and
adders SM0 to SM5, which are connected in series with each other, and scale multipliers
a to f, A to E, and &agr; to &egr; realize a configuration corresponding to
the audio filter section 2, the adder 3, and the noise shaping section 4 as shown
in Fig. 2A. The digital audio filter 1 is also configured to include a comparator
CP that corresponds to the quantizer section 5 shown in Fig. 2A.

Then, the digital audio filter 1 can define the filter
characteristic for the one-bit input signal X by mainly adjusting respective coefficient
values of the scale multipliers a to f and A to E, and can confine noise components
such as quantization noise within a frequency band higher than the audio band by
mainly adjusting respective coefficient values of the scale multipliers &agr;
to &egr;. That is, the digital audio filter 1 is adapted to output a noise-shaped
one-bit output signal Y from the comparator CP. Accordingly, it is possible to eliminate
noise components such as quantization noise by passing the one-bit output signal
Y through a low-pass filter which has a high-band cutoff frequency at the upper
limit of the audio band.

According to the conventional digital audio filter 1 configured
as above, the scale multipliers a to f shown in Fig. 1 only have to multiply the
one-bit input signal X of a one-bit data train by a coefficient value of multiple-bit
(hereinafter referred to as "multi-bit") data. Likewise, the scale multipliers A
to E also only have to multiply the one-bit output signal Y of a one-bit data train
by a coefficient value of multi-bit data. Thus, this eliminates the need to form
each of the scale multipliers a to f and A to E using a multiplier that multiplies
multi-bit data by multi-bit data, thereby making it possible to reduce components
and simplify the overall configuration.

Then, for example, the digital audio filter 1 performs
digital filtering on a one-bit input signal X of a one-bit stream that has been
read on a storage medium such as a CD (Compact Disc) or a one-bit input signal X
obtained by being converted from analog to digital by a one-bit A/D converter that
employs the Delta-Sigma modulation scheme. The digital audio filter 1 also outputs
a one-bit output signal Y that has been re-quantized by the comparator CP. This
makes it possible to construct a digital audio signal processing system which allows
the input and output signals X and Y to remain in the form of one-bit stream during
their processing.

[Non-Patent Document 1]
N. M. CASEY and JAMES A. S. ANGUS. "One Bit Digital Processing Audio Signals"
Proc. Audio Eng. 95th AES Convention 1993, New York

__DISCLOSURE OF THE INVENTION__
__PROBLEMS TO BE SOLVED BY THE INVENTION__
Meanwhile, as described above, the conventional digital
audio filter 1 shown in Fig. 1 has an improved configuration to provide the filter
characteristics corresponding to those of such a typical digital audio filter as
shown in Fig. 2A, and is thus represented by the transfer function of the typical
digital audio filter.

That is, assuming that the one-bit input signal X is X(z)
on the z-plane, the one-bit output signal Y is Y(z), the audio filter section 2
shown in Fig. 2A is a transfer function G(z), the noise shaping section 4 is a transfer
function H(z) such as of a low-pass filter, and quantization noise caused in the
quantizer section 5 is Q(z), the relation of the one-bit output signal Y(z) to the
one-bit input signal X(z) can be expressed by Equation (1) below.

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}1}\\ \mathrm{Y}\left(z\right)=\frac{\mathrm{G}\left(z\right)\mathrm{H}\left(z\right)\mathrm{X}\left(z\right)}{1+\mathrm{H}\left(z\right)}+\frac{\mathrm{Q}\left(z\right)}{1+\mathrm{H}\left(z\right)}\end{array}$$

Here, Equation (1) above shows that the transfer function
for the one-bit input signal X(z) at the first term on the right-hand side is expressed
by the multiplicative and divisional relation between the transfer function G(z)
of the audio filter section 2 and the transfer function H(z) of the noise shaping
section 4. It is also shown that the transfer function for the quantization noise
Q(z) at the second term on the right-hand side is expressed by the divisional relation
with the transfer function H(z) of the noise shaping section 4.

These relations thus show that an adjustment or the like
made to the transfer function H(z) of the noise shaping section 4 at the second
term would cause the transfer function G(z) of the audio filter section 2 to be
affected by the transfer function H(z) and thus substantially changed, thereby resulting
in a change in the audio filter characteristic for the one-bit input signal X(z).

Furthermore, the digital audio filter 1 has a feature that
since the scale multipliers a to f, A to E, and &agr; to &egr; shown in Fig.
1 are provided in order to realize the transfer functions of the first term and
the second term, each coefficient value of these scale multipliers a to f, A to
E, and &agr; to &egr; has complicated effects on both the transfer function
G(z) of the audio filter section 2 and the transfer function H(z) of the noise shaping
section 4.

Accordingly, in practice, separate adjustments made to
the coefficient values of the scale multipliers a to f and A to E and the coefficient
values of the scale multipliers &agr; to &egr; would make it difficult to define
desired characteristics for each filter characteristic (e.g., gain and phase versus
frequency characteristics) of the audio filter section 2 and the noise shaping section
4. It is thus required to make total adjustments in consideration of the effects
exerted between all the coefficient values of the scale multipliers a to f, A to
E, and &agr; to &egr;, thereby causing adjustments or design works to be significantly
complicated.

More specifically, as schematically shown in Fig. 2B, an
attempt was made to design the noise shaping section 4 which provided a desired
low-band rejection characteristic for a noise component such as a quantization error.
In this case, there was a problem that an adjustment made to the transfer function
H(z) of the noise shaping section 4, e.g., to change the low-band rejection characteristic
caused the resulting transfer function H(z) to have effects on the transfer function
G(z) of the audio filter section 2 from the relation given by Equation (1) above.
For example, this resulted in changes in the cutoff frequency or the cutoff characteristic
of the audio filter section 2 and the noise shaping section 4, thereby adversely
affecting the one-bit input signal X.

An attempt was also made to define a desired filter characteristic
corresponding to the transfer functions G(z) and H(z) by performing computer simulations
to determine the coefficient values of the scale multipliers a to f, A to E, and
&agr; to &egr;. In this case, there were problems that to optimize these many
coefficient values that simultaneously affected both the transfer functions G(z)
and H(z), enormous and long-duration processing was required, while to realize the
digital audio filter 1 which had a stable frequency versus gain and phase relation,
more enormous and longer-duration processing was required.

The present invention was developed in view of such conventional
problems. It is an object of the present invention to provide a digital filter which
allows for separately and independently adjusting and designing an intended filter
characteristic and a filter characteristic for noise shaping, and a method for designing
the same.

It is also an object of the present invention to provide
a simply configured digital filter which allows for separately and independently
adjusting and designing an intended filter characteristic and a filter characteristic
for noise shaping, and a method for designing the same.

__MEANS FOR SOLVING THE PROBLEMS__
The invention as set forth in claim 1 is to provide a digital
filter for applying an intended filter characteristic to an input signal sampled
at a frequency higher than the Nyquist frequency for output. The digital filter
is characterized by comprising: main filter means, having a transfer function which
provides an intended filter characteristic, for filtering an input signal based
on the transfer function; quantization means for re-quantizing an output from the
main filter means to output an output signal; sub-filter means, having an inverse
transfer function from the transfer function of the main filter means, for filtering
the output signal based on the inverse transfer function; noise shaping means, having
a transfer function created by allowing a transfer function of a low-band rejection
filter to be subtracted from 1, for filtering an input signal based on the transfer
function, the low-band rejection filter having a low-band rejection characteristic
in a predetermined frequency band of a noise component including a quantization
error caused at the quantization means; first computation means for adding the input
signal and an output from the noise shaping means to supply the resulting sum signal
to the main filter means; and second computation means for computing the difference
between the sum signal supplied to the main filter means and the output from the
sub-filter means to supply the resulting differential signal to the noise shaping
means.

The invention as set forth in claim 2 is to provide the
digital filter according to claim 1, characterized in that the noise shaping means
is formed of a removably connected unit.

The invention as set forth in claim 3 is to provide the
digital filter according to claim 1 or 2, characterized in that the transfer function
of the low-band rejection filter is configured to be expressed by a transfer function
(z-1)^{n}/(z-p)^{n} on the z-plane, where an arbitrary natural number
n is its order and a coefficient p corresponds to its pole.

The invention as set forth in claim 4 is to provide a digital
filtering method for applying an intended filter characteristic to an input signal
sampled at a frequency higher than the Nyquist frequency for output. The method
is characterized by comprising: a main filter process, having a transfer function
which provides an intended filter characteristic, for filtering an input signal
based on the transfer function; a quantization process for re-quantizing an output
created in the main filter process to output an output signal; a sub-filter process,
having an inverse transfer function from the transfer function in the main filter
process, for filtering the output signal based on the inverse transfer function;
a noise shaping process, having a transfer function created by allowing a transfer
function of a low-band rejection filter to be subtracted from 1, for filtering an
input signal based on the transfer function, the low-band rejection filter having
a low-band rejection characteristic in a predetermined frequency band of a noise
component including a quantization error caused in the quantization process; a first
computation process for adding the input signal and an output created in the noise
shaping process to supply the resulting sum signal to the main filter process; and
a second computation process for computing the difference between the sum signal
supplied to the main filter process and the output produced in the sub-filter process
to supply the resulting differential signal to the noise shaping process.

The invention as set forth in claim 5 is to provide a computer
program which realizes a digital filter for applying an intended filter characteristic
to an input signal sampled at a frequency higher than the Nyquist frequency for
output. The computer program is characterized by comprising: a main filter step,
having a transfer function which provides an intended filter characteristic, for
filtering an input signal based on the transfer function; a quantization step for
re-quantizing an output created in the main filter step to output an output signal;
a sub-filter step, having an inverse transfer function from the transfer function
in the main filter step, for filtering the output signal based on the inverse transfer
function; a noise shaping step, having a transfer function created by allowing a
transfer function of a low-band rejection filter to be subtracted from 1, for filtering
an input signal based on the transfer function, the low-band rejection filter having
a low-band rejection characteristic in a predetermined frequency band of a noise
component including a quantization error caused in the quantization step; a first
computation step for adding the input signal and an output created in the noise
shaping step to supply the resulting sum signal to the main filter step; and a second
computation step for computing the difference between the sum signal supplied to
the main filter step and the output produced in the sub-filter step to supply the
resulting differential signal to the noise shaping step.

The invention as set forth in claim 6 is to provide a method
for designing a digital filter which applies an intended filter characteristic to
an input signal sampled at a frequency higher than the Nyquist frequency for output.
The digital filter includes: main filter means having a transfer function which
provides an intended filter characteristic for an input signal; quantization means
for re-quantizing an output from the main filter means to output an output signal;
sub-filter means for filtering the output signal for output; computation means for
computing the difference between the signal supplied to the main filter means and
the output from the filter means to output the resulting differential signal; noise
shaping means for filtering the differential signal for output; and another computation
means for adding an output from the noise shaping means and the input signal for
delivery to the main filter means. The method is characterized by comprising the
steps of: creating sub-filter means, having an inverse transfer function from the
transfer function of the main filter means, for filtering the output signal based
on the inverse transfer function; and creating a transfer function of the noise
shaping means by appropriately determining a transfer function of a low-band rejection
filter having a low-band rejection characteristic in a frequency band in which a
noise component including a quantization error is caused, and by subtracting the
transfer function of the low-band rejection filter from 1.

The invention as set forth in claim 7 is to provide the
method for designing a digital filter according to claim 6, characterized in that
the noise shaping means is formed of a removably connected unit.

The invention as set forth in claim 8 is to provide the
method for designing a digital filter according to claim 6 or 7, characterized in
that the transfer function of the low-band rejection filter is configured to be
expressed by a transfer function (z-1)^{n}/(z-p)" on the z-plane, where
an arbitrary natural number n is its order and a coefficient p corresponds to its
pole.

The invention as set forth in claim 9 is to provide the
method for designing a digital filter according to claim 8, characterized by further
comprising the steps of: creating an open loop transfer function K(z)/(1-K(z)) based
on a transfer function K(z) obtained by subtracting the transfer function (z-1)^{n}/(z-p)^{n}
of the low-band rejection filter from 1; varying a coefficient p in the open loop
transfer function K(z)/(1-K(z)) as a parameter within the range of from -1 to +1;
determining a coefficient p in the open loop transfer function K(z)/(1-K(z)) with
a phase margin based on frequency versus gain and frequency versus phase relations;
and determining a transfer function of the low-band rejection filter by employing
the resulting coefficient p.

The invention as set forth in claim 10 is to provide the
method for designing a digital filter according to claim 9, characterized by further
comprising the step of determining a transfer function of a low-band rejection filter,
having a low-band rejection characteristic in a frequency band in which a noise
component including the quantization error is caused, based on the frequency versus
gain relation for the determined transfer function of the low-band rejection filter.

__BRIEF DESCRIPTION OF THE DRAWINGS__

- Fig. 1
- is a block diagram illustrating the configuration of a conventional digital
filter.
- Fig. 2A and 2B
- are views illustrating the configuration of a typical digital filter which the
digital filter shown in Fig. 1 is based on.
- Fig. 3
- is a block diagram illustrating the configuration of a digital filter according
to an embodiment of the present invention.
- Fig. 4
- is a block diagram illustrating the configuration of a digital filter according
to a more specific example.
- Fig. 5
- is a characteristic diagram illustrating the frequency versus gain characteristic
of an open loop transfer function OP(z) employed for designing the digital filter
shown in Fig. 4.
- Fig. 6
- is a characteristic diagram illustrating the frequency versus phase characteristic
of the open loop transfer function OP(z) employed for designing the digital filter
shown in Fig. 4.
- Fig. 7
- is a characteristic diagram illustrating the frequency versus gain characteristic
of a low-band rejection filter employed for designing the digital filter shown in
Fig. 4.
- Fig. 8
- is a characteristic diagram illustrating an extracted portion of the frequency
versus gain characteristic of the low-band rejection filter shown in Fig. 7.
- Fig. 9
- is a view illustrating the waveforms of input and output signals assuming that
there is no delay in the open loop transfer function of the digital filter shown
in Fig. 4.
- Fig. 10
- is a view illustrating the waveforms of input and output signals assuming that
there is a delay in the open loop transfer function of the digital filter shown
in Fig. 4.

__BEST MODE FOR CARRYING OUT THE INVENTION__
Now, with reference to Fig. 3, a description will be given
of an embodiment for carrying out the present invention.

Fig. 3 is a block diagram illustrating the configuration of a digital filter according
to this embodiment.

In the drawing, the digital filter 10 is configured to
have an adder 11; a main filter section 12 having a transfer function for applying
an intended filter characteristic to an input signal X; a quantizer section 13;
a sub-filter section 14 having an inverse transfer function from the main filter
section 12; a subtracter 15; and a noise shaping section 16.

The adder 11 adds the input signal X and a feedback signal
D5 output from the noise shaping section 16, and then supplies the resulting sum
signal D1 to the main filter section 12 and the subtracter 15.

The main filter section 12 performs digital filtering on
the sum signal D1 based on the aforementioned transfer function to thereby produce
and output a signal obtained by applying an intended filter characteristic to the
input signal X (hereinafter referred to as the "main filter signal") D2.

That is, the main filter section 12 is a main filter for
applying an intended filter characteristic to the input signal X, and is formed
of a filter that meets requirements such as design specifications. For example,
to make adjustments to the frequency characteristic of the input signal X in various
ways, the main filter section 12 is formed of an equalizer, while to pass the input
signal X therethrough in its frequency band, the main filter section 12 is formed
of a band-pass filter. On the other hand, to pass predetermined high-frequency components
of the input signal X therethrough, it is formed of a high-pass filter, whereas
to pass predetermined low-frequency components of the input signal X therethrough,
it is formed of a low-pass filter. Additionally, the configuration of those filters
can be determined as appropriate.

The quantizer section 13 re-quantizes the main filter signal
D2 and then outputs the re-quantized signal as an output signal Y.

The sub-filter section 14 has an inverse transfer function
from the main filter section 12, and outputs a signal (hereinafter referred to as
the "sub-filter signal") D3 which is obtained by performing digital filtering on
the output signal Y based on the inverse transfer function.

That is, with the transfer function of the main filter
section 12 represented by G(z) on the z-plane, the sub-filter section 14 is formed
of a filter having the inverse transfer function G^{-1}(z) from the transfer
function G(z). The sub-filter section 14 performs digital filtering on the output
signal Y(z) based on the transfer function G-^{1}(z) to thereby output the
sub-filter signal D3(z) represented by Y(z) G^{-1}(z).

The subtracter 15 computes the difference between the sum
signal D 1 and the sub-filter signal D3, and then supplies the resulting differential
signal D4 to the noise shaping section 16.

The noise shaping section 16 has a predetermined transfer
function K(z), described later, and performs computations on the differential signal
D4 based upon the transfer function K(z) to thereby create the feedback signal D5,
which is in turn fed back to the adder 11 to be added to the input signal X. Note
that the transfer function K(z) of the noise shaping section 16 will be described
later in more detail.

Then, in the digital filter 10 configured as such, suppose
that an input signal X which has been oversampled at a frequency fs higher than
the Nyquist frequency is supplied to the adder 11. In this case, the adder 11, the
main filter section 12, the quantizer section 13, the sub-filter section 14, the
subtracter 15, and the noise shaping section 16 perform digital computational operations
on it in sync with the sampling cycle Ts (the reciprocal of the frequency fs) of
the oversampling and perform the intended filtering on the input signal X based
on the transfer function of the main filter section 12, allowing a re-quantized
output signal Y to be delivered from the quantizer section 13.

Furthermore, the noise shaping section 16 provides a so-called
noise shaping function, thereby allowing the quantizer section 13 to output a noise-shaped
output signal Y.

That is, referring to Fig. 2B for description, suppose
that the main filter section 12 defines, e.g., a filter characteristic having the
audio band as a pass band, and the noise shaping section 16 defines a filter characteristic
having a desired low-band rejection characteristic. In this case, the quantizer
section 13 outputs the output signal Y which will cause a noise component such as
a quantization error to be confined in a frequency band higher than the audio band.
Accordingly, passing the output signal Y through a low-pass filter or the like having
a high-band cutoff frequency at the upper limit of the audio band makes it possible
to create an audio signal whose noise components such as a quantization error have
been eliminated.

A description will now be made to the feature of the digital
filter 10 according to this embodiment.

The output signal Y(z) is expressed by Equation (2) below and the feedback signal
D5(z) is expressed by Equation (3) below, where the input signal X is represented
by X(z) on the z-plane, the output signal Y by Y(z), the feedback signal D5 delivered
from the noise shaping section 16 by D5(z), the transfer function of the main filter
section 12 by G(z), the quantization noise introduced in the quantizer section 13
by Q(z), the transfer function of the sub-filter section 14 by G^{-1}(z),
and the transfer function of the noise shaping section 16 by K(z).

Accordingly, combining Equations (2) and (3) below together
to eliminate the term of the feedback signal D5(z) gives the relation of the output
signal Y(z) to the input signal X(z), i.e., the transfer function of the digital
filter 10 expressed by Equation (4) below.

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}2}\\ \mathrm{Y}\left(z\right)=\left\{\mathrm{X},\left(z\right),+,\mathrm{D},,5,\left(z\right)\right\}\mathrm{G}\left(z\right)+\mathrm{Q}\left(z\right)\end{array}$$

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}3}\\ \mathrm{D}5\left(z\right)=\left\{\mathrm{X},\left(z\right),+,\mathrm{D},,5,\left(z\right),-,\mathrm{Y},\left(z\right),,{\mathrm{G}}^{-\mathit{1}},\left(z\right)\right\}\mathrm{K}\left(z\right)\end{array}$$

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}4}\\ \mathrm{Y}\left(z\right)=\mathrm{G}\left(z\right)\mathrm{X}\left(z\right)+\left\{1,-,\mathrm{K},\left(z\right)\right\}\mathrm{Q}\left(z\right)\end{array}$$

Here, as can be seen in the first term on the right-hand
side of Equation (4) above, the input signal X(z) is multiplied only by the transfer
function G(z) of the main filter section 12, whereas as also can be seen in the
second term on the right-hand side, the quantization noise Q(z) is multiplied by
the transfer function K(z) of the noise shaping section 16.

That is, the transfer function G(z) of the main filter
section 12 that is used for applying a desired filter characteristic to the input
signal X(z) is clearly separated from the transfer function K(z) of the noise shaping
section 16 that is used for noise shaping of noise components such as the quantization
noise Q(z). Accordingly, this makes it possible to separately and independently
adjust and design the main filter section 12 and the noise shaping section 16.

Then, in determining the transfer function K(z) of the
noise shaping section 16, a low-band rejection filter is designed, as appropriate,
which has a low-band rejection characteristic in a frequency band in which the quantization
noise Q(z) to be eliminated is caused. Then, using the transfer function W(z) of
the designed low-band rejection filter, based on the relation of Equation (5) below,
a transfer function (1-W(z)) obtained by subtracting the transfer function W(z)
from 1 is employed as the transfer function K(z) of the noise shaping section 16.
Only this procedure makes it possible to readily design the noise shaping section
16.

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}5}\\ \mathrm{W}\left(z\right)=1-\mathrm{K}\left(z\right)\end{array}$$

As such, the digital filter 10 of this embodiment is configured
to allow the transfer function G(z) of the main filter section 12 and the transfer
function K(z) of the noise shaping section 16 to be separately and independently
designed. Therefore, this eliminates the need for considering effects that parameters
such as orders and coefficients of each of the transfer function G(z) and K(z) have
therebetween. Accordingly, this eliminates the need for such design works as described
in the Background Art, and makes it possible to readily design a digital filter
having an intended filter characteristic.

On the other hand, as expressed by Equation (1) above,
the digital filter 1 of the conventional technique allows the transfer function
H(z) of the noise shaping section 4 and the transfer function G(z) of the audio
filter section not to be independently separated from each other. Therefore, although
a digital filter having a fixed filter characteristic could be created, it was difficult
to realize a so-called equalizer or the like for appropriately making gain adjustments
or the like to the input signal X.

In contrast to this, the digital filter 10 of this example
is configured such that the transfer function G(z) of the main filter section 12
and the transfer function G^{-1}(z) of the sub-filter section 14 are independent
of the transfer function K(z) of the noise shaping section 16. Therefore, an adjustment
made only to the transfer functions G(z) and G^{-1}(z) allows for providing
desired equalizer characteristics. From this, the digital filter 10 of this example
provides a good function as a so-called variable digital filter which is not only
easy to design but also capable of making adjustments as appropriate to its filter
characteristic even after it is completely assembled.

Furthermore, the digital filter 10 of this example is configured
such that the transfer function G(z) of the main filter section 12 and the transfer
function K(z) of the noise shaping section 16 are independently related to each
other. Thus, the digital filter 10 provides a good function which allows for altering
the noise shaping characteristic for noise components such as quantization noise
only by changing the transfer function K(z) of the noise shaping section 16.

Furthermore, in the foregoing description, it was not made
clear that the input signal X and the output signal Y were either a one-bit stream
signal of PWM modulated waves or PDM modulated waves or a signal of a multi-bit
data train. However, the digital filter 10 of this embodiment is applicable even
to the cases where both the input signal X and the output signal Y are a multi-bit
data signal or one-bit stream signal, or one of the input signal X and the output
signal Y is a multi-bit data signal and the other is a one-bit stream signal.

That is, in the internal configuration of each of the adder
11, the main filter section 12, the quantizer section 13, the sub-filter section
14, the subtracter section 15, and the noise shaping section 16, such a portion
of the configuration that allows for performing computational operations on a one-bit
stream signal as it is can be configured to compute one-bit stream signals. On the
other hand, such a portion of the configuration that requires multiplications or
the like between pieces of multi-bit data may be configured to perform computational
operations on data of a long bit length so as not to cause so-called rounding-off
errors or the like.

Accordingly, the digital filter 10 of this embodiment is
applicable regardless of whether the input signal X and the output signal Y are
either a one-bit stream signal of PWM modulated waves or PDM modulated waves, or
multi-bit data.

Additionally, the digital filter 10 of this embodiment
can be implemented with hardware circuitry that employs so-called discrete circuits
or integrated circuit devices (IC) as well as with digital signal processors (DSP)
or micro-computer systems, which are adapted for digital computations.

A description will now be given of a method for designing
the digital filter 10 of this embodiment.

Note that the description will be specifically directed
to a method for designing the digital filter 10 using a computer system.

To begin with, the main filter section 12 shown in Fig.
3 is designed which has an intended transfer function G(z).

Then, from the transfer function G(z) of the main filter section 12, the sub-filter
section 14 having the transfer function G^{-1}(z) is designed.

Then, a low-band rejection filter is designed which corresponds
to the transfer function (1-K(z)) shown in the second term on the right-hand side
of Equation (4) above. In other words, for example, the low-band rejection filter
which provides such a low-band rejection characteristic as shown in Fig. 2B may
be implemented as follows. That is, the configuration of a filter which can provide
the low-band rejection characteristic is determined as appropriate, and then simulations
are performed to optimize parameters such as the order and coefficients of the filter,
thereby designing a low-band rejection filter which provides the intended low-band
rejection characteristic.

Then, based on the relation given by Equation (5) above,
the transfer function W(z) of the resulting low-band rejection filter is subtracted
from 1, thereby determining the transfer function K(z) of the noise shaping section
16, i.e., the transfer function K(z) that is 1-W(z).

Then, as shown in Fig. 3, based on the transfer functions
G(z), G^{-1}(z), and K(z) which are determined through the processes described
above, the main filter section 12, the sub-filter section 14, and the noise shaping
section 16 are actually formed with electronic circuits. In addition, wiring is
carried out therebetween including the adder 11, the quantizer section 13, and the
subtracter 15, thus providing the digital filter 10 of this embodiment completed
as such.

On the other hand, a digital signal processor (DSP) or
micro-computer system may be employed to implement the digital filter 10 of this
embodiment, or a so-called running program for a micro-computer system or the like
to execute may be implemented. To this end, based on the transfer functions G(z),
G^{-1}(z), and K(z) which have been determined, a computer program corresponding
to the block configuration shown in Fig. 3 is created.

As can be seen from above, the digital filter 10 of this
embodiment allows for separately and independently designing the transfer function
G(z) of the main filter section 12 and the transfer function K(z) of the noise shaping
section 16, thereby making its design method simplified and very easy to create.

Additionally, in the design method described above, after
the transfer function G(z) of the main filter section 12 and the transfer function
G^{-1}(z) of the sub-filter section 14 have been determined, then the transfer
function K(z) of the noise shaping section 16 is determined. However, since the
digital filter 10 of this embodiment can be advantageously configured to allow for
separately and independently designing the transfer functions G(z) and K(z), the
order of designing these transfer functions G(z), G^{-1}(z), and K(z) is
not important, but they can be designed in an appropriate order.

On the other hand, to design a digital filter which is
used for electronic devices, the transfer function K(z) of the noise shaping section
16 can be first designed. This provides an effect that the transfer function K(z)
of the noise shaping section 16 needs not to be re-designed even when a change or
an adjustment is subsequently made to the characteristics of the main filter section
12 and the sub-filter section 14.

Furthermore, since the transfer function K(z) of the noise
shaping section 16 needs not to be re-designed, the noise shaping section 16 may
be formed separately with an integrated circuit device (IC) or the like, or formed
by so-called unitization. Then, when various digital filters with the main filter
section 12 having different filter characteristics are formed, the unitized noise
shaping section 16 may be connected between the adder 11 and the subtracter 15.

As described above, the noise shaping section 16 unitized
in advance can be provided as a versatile electronic component for designing digital
filters.

__Example__
A description will now be given of a more specific example
with reference to Fig. 4 to Fig. 10. Note that Fig. 4 is a block diagram illustrating
the configuration of an audio digital filter of this example, with those portions
similar or corresponding to those of Fig. 3 being indicated with the same symbols.

Additionally, the digital filter 10 is a so-called audio
graphic equalizer which is configured to receive a one-bit stream digital audio
signal (one-bit input signal) X that is read on a storage medium such as a CD or
DVD (Digital Versatile Disc) and apply an intended filter characteristic thereto,
then re-quantizing the resulting signal to output a one-bit stream one-bit output
signal Y. The digital filter 10 is applicable to CD players, DVD players, or digital
audio devices which allow the speaker to sound via a D-class amplifier or the like.

Description is now made to the configuration of the digital
filter 10 with reference to Fig. 4. Like the digital filter shown in Fig. 3, the
digital filter 10 is configured to have the adder 11, the main filter section 12,
the quantizer section 13, the sub-filter section 14, the subtracter 15, and the
noise shaping section 16.

Here, the adder 11 adds a one-bit input signal X and a
multi-bit feedback signal D5 from the noise shaping section 16, and then outputs
a multi-bit sum signal D1. As such, the adder 11 only adds the one-bit stream signal
and the multi-bit signal, and thus can be implemented in a simple configuration,
thereby contributing to the simplification of the configuration of the digital filter
10.

The main filter section 12, which has the transfer function
G(z) expressed by Equation (6) below to apply an intended filter characteristic
to the one-bit input signal X, receives the sum signal D 1 to output the main filter
signal D2.

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}6}\\ \mathrm{G}\left(z\right)=\frac{{\mathrm{a}}_{0}{z}^{2}+{\mathrm{a}}_{1}z+{\mathrm{a}}_{2}}{{z}^{2}-{\mathrm{b}}_{1}z-{\mathrm{b}}_{2}}\end{array}$$

Then, coefficients a_{0}, a_{1}, a_{2}, b_{1}, and
b_{2} are set to predetermined coefficient values, thereby realizing a graphic
equalizer which has the audio band as a pass band.

Note that this example employs a graphic equalizer as the
main filter section 12; however, a low-pass filter, a band-pass filter, a high-pass
filter, or other various types of filters may also be employed according to design
specifications or the like.

The quantizer section 13, which is formed of a comparator
for comparing a predetermined threshold THD with the main filter signal D2, outputs
a one-bit output signal Y of a one-bit stream which has binary levels of logic "1"
or "0" according to the magnitude of the level of the main filter signal D2 with
respect to the threshold THD.

The sub-filter section 14, which has the inverse transfer
function G^{-1}(z) from the transfer function G(z), performs filtering on
the one-bit output signal Y to output the sub-filter signal D3. That is, the sub-filter
section 14 has the transfer function G^{-1}(z)_expressed by Equation (7)
below.

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}7}\\ \mathrm{G}\left(z\right)=\frac{{z}^{2}-{\mathrm{b}}_{1}z-{\mathrm{b}}_{2}}{{\mathrm{a}}_{0}{z}^{2}+{\mathrm{a}}_{1}z+{\mathrm{a}}_{2}}\end{array}$$

Here, the coefficients b_{1}, b_{2}, a_{0h},
a_{1h}, and a_{2h} in Equation (7) above are coefficient values
which occur when the inverse transfer function G^{-1}(z) is derived from
the transfer function G(z).

The subtracter 15 subtracts the sub-filter signal D3 from
the sum signal D1, and then supplies the resulting differential signal D3 to the
noise shaping section 16.

The noise shaping section 16, which has the same transfer
function K(z) as the one described in relation to Equation (5) above, performs filtering
on the differential signal D3 based on the transfer function K(z), thereby creating
a feedback signal D1 to be supplied to the adder 11.

Here, the transfer function K(z) is determined as follows.

To begin with, as described in the embodiment, the transfer function W(z) of the
low-band rejection filter is the transfer function of the output signal Y(z) to
the quantization noise Q(z), which is expressed by subtracting the transfer function
K(z) from 1. The transfer function W(z) of the low-band rejection filter is determined
as a simple transfer function that is expressed by Equation (8) below.

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}8}\\ \mathrm{W}\left(z\right)=1-\mathrm{K}\left(z\right)=\frac{{\left(z,-,1\right)}^{\mathrm{n}}}{{\left(z,-,p\right)}^{\mathrm{n}}}\end{array}$$

Then, the order n of the transfer function W(z) is determined
so that the cutoff characteristic at a cutoff frequency for eliminating quantization
noise or the like conforms to the requirements of the design specification or the
like. For example, to provide a steep cutoff characteristic, the order n is increased.
Accordingly, when the low-band rejection filter is designed as a fourth order filter,
its transfer function W(z) is expressed by Equation (9) below.

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}9}\\ \mathrm{W}\left(z\right)=1-\mathrm{K}\left(z\right)=\frac{{\left(z,-,1\right)}^{4}}{{\left(z,-,p\right)}^{4}}\end{array}$$

The coefficient p in Equation (9) above, or a parameter
serving as a pole of the transfer function W(z), is provided with a so-called optimized
value, thereby determining a stable low-band rejection filter which has a phase
margin in terms of a frequency versus gain and phase relation. Then, in the case
of the fourth order low-band rejection filter, once the value of the coefficient
p of the stable transfer function W(z) is determined, the noise shaping section
16 having the transfer function K(z) expressed by Equation (10) below is determined
from the relation given by Equation (9) above.

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}10}\\ \mathrm{K}\left(z\right)=\frac{\left(-,4,,p,+,4\right){z}^{3}+\left(6,,{p}^{2},-,6\right){z}^{2}+\left(-,4,,{p}^{3},+,4\right)z+\left({p}^{4},-,1\right)}{{\left(z,-,1\right)}^{4}}\end{array}$$

In the digital filter 10 configured as described above,
the one-bit input signal X which is oversampled at a the frequency fs higher than
the Nyquist frequency may be supplied to the adder 11. In this case, the adder 11,
the main filter section 12, the quantizer section 13, the sub-filter section 14,
the subtracter 15, and the noise shaping section 16 perform digital computational
operations on it in sync with the sampling cycle Ts (the reciprocal of the frequency
fs) of the oversampling and perform filtering on the one-bit input signal X based
on the transfer function G(z) of the main filter section 12, allowing a re-quantized
one-bit output signal Y to be delivered from the quantizer section 13. The quantizer
section 13 then outputs a one-bit output signal Y which is noise shaped by the function
of the noise shaping section 16.

A description will now be given of a method for designing
the digital filter 10 of this example.

Note that the description will be directed to a method for designing the digital
filter 10 using a computer system. Furthermore, for convenience of the explanation,
the description is directed to a case where the noise shaping section 16 is designed
from the fourth order low-band rejection filter expressed by Equation (9) above.

To begin with, the main filter section 12 having the transfer
function G(z) shown in Fig. 4 is designed, and then the inverse transfer function
G^{-1}(z) of the main filter section 12 is employed as the transfer function
of the sub-filter section 14. The transfer function G^{-1}(z) shown in Equation
(7) above is thus determined.

Then, the transfer function W(z) of the simple low-band
rejection filter shown in Equation (9) above is designed by simulation.

Here, to design the transfer function W(z) which provides
good stability, the optimum value of the coefficient p of an open loop transfer
function OP(z) expressed by Equation (11) below is analyzed and determined by simulation.

For example, with the main filter section 12 being disconnected
from the quantizer 13, the transfer function OP(z) of the open loop from the input
terminal of the quantizer 13 to the output terminal of the main filter section 12
is simulated.

$$\begin{array}{l}\phantom{\rule{7em}{0ex}}\underset{\_}{\mathrm{Equation}\phantom{\rule{1em}{0ex}}11}\\ \begin{array}{ll}\mathrm{OP}\left(z\right)& =\frac{\mathrm{K}\left(z\right)}{1-\mathrm{K}\left(z\right)}\\ \phantom{\rule{1em}{0ex}}& =\frac{\left(-,4,,p,+,4\right){z}^{3}+\left(6,,{p}^{2},-,6\right){z}^{2}+\left(-,4,,{p}^{3},+,4\right)\left({p}^{4},-,1\right)z}{{\left(z,-,1\right)}^{4}}\end{array}\end{array}$$

Then, the coefficient p of the open loop transfer function
OP(z) expressed by Equation (11) above is varied within the range of from -1 to
+1 to analyze the frequency versus gain and phase characteristic by simulation.
Varying the coefficient p within the range of from real number -1 to +1 in this
manner corresponds to varying the coefficient p associated with a pole within the
range (the stable range) of a unit circle on the z-plane, and is thus preferred
means for positively deriving a stable transfer function W(z).

As can be seen from the results of analysis obtained by
such a simulation and allowed to appear on a display or the like, the frequency
versus gain characteristic of the transfer function OP(z) as shown in Fig. 5 is
obtained which employs the coefficient p as a parameter. Additionally, the frequency
versus phase characteristic of the transfer function OP(z) as shown in Fig. 6 is
obtained which employs the coefficient p as a parameter. Furthermore, the frequency
versus gain characteristic of the transfer function W(z) of the low-band rejection
filter is simulated while the coefficient p is being varied within the range of
from real number -1 to +1, and is then allowed to appear on a display or the like.
The results shown in Fig. 7 are thus obtained.

In this context, the frequency versus gain characteristic
(see Fig. 5) and the frequency versus phase characteristic (see Fig. 6) of the transfer
function OP(z), and the frequency versus gain characteristic (see Fig. 7) of the
transfer function W(z) are analyzed. Then, the coefficient p is first selected under
the conditions in which the transfer function OP(z) of Figs. 5 and 6 provides a
sufficient phase margin with stability, and thereafter, a check is made to determine
whether the frequency versus gain characteristic is representative of the desired
low-band rejection characteristic when the selected coefficient p is applied to
the transfer function W(z). In this manner, the transfer function W(z) which is
stable and indicative of the desired low-band rejection characteristic is determined.

In this regard, as is clear from the characteristics in
Figs. 5, 6, and 7, it can be seen that the value of the coefficient p being 0.7
can make it possible to realize a low-band rejection filter which has a sufficient
phase margin and satisfies the condition of the intended frequency versus gain characteristic.

Fig. 8 is a view illustrating a frequency versus gain characteristic
extracted from the characteristics of the transfer function W(z) of the low-band
rejection filter shown in Fig. 7, with the value of the coefficient p being 0.7.
Only by checking if such a frequency versus gain characteristic conforms to the
intended low-band rejection characteristic, the transfer function W(z) of the low-band
rejection filter can be easily determined.

Next, using the transfer function W(z) of the low-band
rejection filter, the transfer function K(z) of the noise shaping section 16 is
derived from the relation given by Equation (9) above, thereby determining the transfer
function K(z) shown in Equation (10) above.

Then, based on the transfer functions G(z), G^{-1}(z),
and K(z) which are determined through the processes described above, the digital
filter 10 shown in Fig. 4 is formed, and thus the design work of the digital filter
10 is ended.

A description will now be given of the features of the
digital filter 10 of this example.

Like the digital filter of the embodiment shown in Fig. 3, the digital filter 10
of this example is configured to allow for separately and independently designing
the transfer function G(z) of the main filter section 12 and the transfer function
K(z) of the noise shaping section 16. It is thus possible to adjust or design each
of the coefficients a_{0}, a_{1}, a_{2}, b_{1},
b_{2}, a_{0h}, a_{1h}, and a_{2h} shown in Equations
(7) and (8) above separately and independently of the noise shaping section 16.
This makes it possible to easily and accurately design a digital filter having an
intended filter characteristic.

Furthermore, the transfer function W(z) of the low-band
rejection filter that is used to derive the transfer function K(z) of the noise
shaping section 16 is given a simple configuration with a less number of parameters
shown in Equation (8) above (i.e., the coefficient p and order n). Therefore, in
checking or analyzing the stability of the phase margin or the like of the digital
filter 10, a simulation can be easily performed and allowed to appear on a display
or the like to thereby easily conduct the analysis or the like, thus facilitating
design works or the like.

The digital filter 10 of this example also ensures that
the main filter section 12 and the sub-filter section 14 are independent of the
noise shaping section 16. Thus, the noise shaping section 16 can be separately formed
with an integrated circuit device (IC) or the like, or formed by so-called unitization,
so as to be connected between the adder 11 and the subtracter 15.

Additionally, the digital filter 10 of this example is
capable of processing both the input and output signals X and Y as a one-bit stream
signal, and can be thus employed for a digital system that uses the Delta-Sigma
Modulation scheme, thereby advantageously providing improved SN ratios.

Additionally, from the results of analysis to be described
below, the digital filter 10 of this example is suitably formed using a digital
signal processor (DSP).

That is, as shown in Fig. 9, assuming that there is no
time delay in the open loop transfer function of the digital filter 10, an output
signal to an input signal was simulated to allow each signal to appear on a display
as an analog signal, for convenience sake.

As shown in Fig. 10, assuming that there exists a delay
time of one hundredth of the sampling cycle Ts in the open loop transfer function
of the digital filter 10, an output signal was simulated when an input signal was
received to allow each signal to appear on a display as an analog signal, for convenience
sake.

A comparison between the waveforms of each signal in Fig.
9 and Fig. 10 shows that even with a delay time of 0.01 Ts, the waveform of the
output signal during the transient response to the input signal is stable without
any distortion, while the waveform of the output signal is also stable without any
distortion after a transition from the transient response period to the steady state.

As a result, it can be concluded that even though the design
method of this example is a simple one, the method allows for designing the digital
filter 10 to have a sufficient phase margin, so that even in the presence of a delay
time in a closed loop, the digital filter 10 would never be unstable and provide
generally the same response characteristic as in the absence of a delay.

This means that the digital filter 10 of this example is
suitably implemented in a digital signal processor (DSP) that takes much time to
perform digital operations.

Furthermore, the effect that the digital filter 10 of this
example is suitably implemented in a digital signal processor enables it to be used
in a wide variety of electronic devices.

For example, the digital filter can be used not only for
CD players or DVD players but also in the field of telecommunications such as for
digital filters which perform digital filtering on intermediate frequency signals,
demodulated signals, or detected signals in a digital tuner that receives digital
broadcast.

Additionally, not only the digital filter 10 of this example
but also the digital filter of the aforementioned embodiment are applicable to the
field of telecommunications as well as to the field of automatic control.

Additionally, the digital filter 10 described in the embodiment
and the example is adapted to keep the main filter section 12 and the sub-filter
section 14 independent of the noise shaping section 16, and is thus also available
as a graphic equalizer with each coefficient of the main filter section 12 and the
sub-filter section 14 being variable.

As described above, the digital filter 10 described in
the embodiment and the example has a very good versatility and is applicable to
a wide variety of digital signal processing systems.